Introduction
Linear equations are one of the most used equations in algebra. In this webquest, you will learn how to solve these linear equations. The general form of a linear equation is slope-intercept form. This form is written as y=mx+b. The y is simply a y coordinate that is on the line of a linear equation. The x is an x coordinate that is shared with the y coordinate. The way these coordinates are shared is in the form of an ordered pair. (x,y) The m in the equation is the slope of the line, or the rate of change. The equation to find the slope of two points is y2-y1/x2-x1. Lastly, the b in slope-intercept is the y-intercept of the equation. The y-intercept is the point of the line where it will cross over the y-axis. The way you find the y-intercept of an equation, given two points, is as follows.
ex. (2,0) (0,-2). First, we must find the slope of the line using the slope formula y2-y1/x2-x1. The first ordered pair given is our x1 and y1, the second point is our x2 and y2.
-2-0/0-2 = -2/-2 = 1 The slope of our line is 1.
Next, we will plug in our value of 1 for m and one of our ordered pairs for x and y.
So, 0=1(2)+b. We will now follow the order of operations. Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction.
We can also remember this as PEMDAS.
First we multiply 1 and 2 giving us 0=2+b.
Then we must subtract the 2 to the other side of the equation giving us -2=b
Now that we have our b and our m, we can plug them in for our final answer.
y=x-2 or y=1x-2
Notice that we can write the x with or without the consonant (number) because it is only 1.
Task
Solve the following linear equations for either one point, the slope, or the y-intercept.
1) Find the slope and y-intercept given the following coordinates.
a. (1,2) (3,-1)
b. (-2,2) (4,-3)
c. (0,4) (4,-3)
2) Find the y-intercept given the slope and one coordinate.
a. m=3 (-1,4)
b. m=1/2 (2,-2)
c. m=-2 (0,5)
3) Find the slope given the y-intercept and one coordinate.
a. b=4 (3,-2)
b. b=2 (2,0)
c. b=0 (2,1)
Process
Solve the following linear equations for either one point, the slope, or the y-intercept.
1) Find the slope and y-intercept given the following coordinates. - For these problems, you must first find the slope of the equation given the two points. y2-y1/x2-x1 Next you will plug in one coordinate given and the slope to find the y-intercept, solving for b in y=mx+b. Lastly, you must enter the slope and y-intercept in slope-intercept form.
a. (1,2) (3,-1)
b. (-2,2) (4,-3)
c. (0,4) (4,-3)
2) Find the y-intercept given the slope and one coordinate. - For these problems, you must plug in the slope and the coordinate to find the y-intercept, solving for b in y=mx+b. Then, you must enter the slope and y-intercept in slope-intercept form.
a. m=3 (-1,4)
b. m=1/2 (2,-2)
c. m=-2 (0,5)
3) Find the slope given the y-intercept and one coordinate. - For these problems, you must plug in the y-intercept and coordinate provided to find the slope, solving for m in y=mx+b. Then, you must enter the slope and y-intercept in slope-intercept form.
a. b=4 (3,-2)
b. b=2 (2,0)
c. b=0 (2,1)
Evaluation
Solve the following linear equations for either one point, the slope, or the y-intercept.
1) Find the slope and y-intercept given the following coordinates.
a. (1,2) (3,-1)
- solution - y= -3/2x+1/2
b. (-2,2) (4,-3)
- solution - y=-5/6x+1/3
c. (0,4) (4,-3)
- solution - y=-7/4x+4
2) Find the y-intercept given the slope and one coordinate.
a. m=3 (-1,4)
- solution - y=3x+y
b. m=1/2 (2,-2)
- solution - y=1/2x-3
c. m=-2 (0,5)
- solution - y=2x+5
3) Find the slope given the y-intercept and one coordinate.
a. b=4 (3,-2)
- solution - y=-2x+4
b. b=2 (2,0)
- solution - y=-x+2
c. b=0 (2,1)
- solution - y=1/2x
Conclusion
Linear equations are used all throughout math, even into higher levels of classes. It is important to have a concrete understanding of them in order to master any mathematics class.
Credits
Myself, using my knowledge of linear equations as a mathematics major to make a webquest of problems off the top of my head.